Invariant sets and the blow up threshold for a nonlocal equation of parabolic type

Liouville theorems and classification results for a nonlocal Schrödinger equation

“…A more general form is the Choquard type equation studied in many papers (such as [14], [22], [31], and [38]), which arises in the study of boson stars and other physical phenomena, and also appears as a continuous-limit 5352 YUTIAN LEI model for mesoscopic molecular structures in chemistry. More related mathematical and physical background can be found in [15], [34], [40] and the references therein. The nonlocal term in (1.1) appears in the example 3.2.8 of the book [4], and it is also related to a simplified model of the Schrödinger-Poisson system (cf.…”

Vacuum isolating, blow up threshold, and asymptotic behavior of solutions for a nonlocal parabolic equation

“…Let δ 1 , δ 2 (δ 1 < δ 2 ) be the two roots of equation d(δ) = J(u 0 ). From Proposition 10 in [21], we know J(u(t)) < d, I(u(t)) > 0 for all t > 0, provided J(u 0 ) < d, I(u 0 ) > 0. Since J(u(t)) ≤ J(u 0 ) < d, I(u(t)) > 0 for each t ≥ 0, we obtain I δ (u(t)) > 0 for δ ∈ (δ 1 , δ 2 ), t ≥ 0 by using Lemma 2.5.…”

“…Precisely, Theorem 1.1. [Theorem 6 and 7 in [21]] Let u 0 ∈ L q (Ω), n − 1 ≤ q < ∞, q > n 2 (p−1)(2− 1 p ). Then there exists T max = T (||u 0 || q ) > 0 such that problem (1.1) possesses a unique classical L q −solution in [0, T max ).…”

“…By using the potential well method, Liu and Ma [21] proved that for low energy solutions (J(u 0 ) < d) the maximum existence time is totally determined by the Nehari functional I(u 0 ). More precisely, if J(u 0 ) < d and I(u 0 ) > 0 then the solution exists globally, if J(u 0 ) < d and I(u 0 ) < 0 then the solution blows up in finite time.…”

“…This paper devoted to continue the study of [21]. The first result of the present paper deals with the solution start with initial data which has low initial energy.…”